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Eider?

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If Eyder is interchangeable with Eider (which I assume it is since you mention the river) then this is what I've found:

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http://www.etymonline.com/index.php?term=eider

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The Danes call the river Ejderen, so changing the j to a y in English makes sense. Old Norse is supposed to be the origin. I'm unsure if eider became duck because of the marshy lands around the Eider River marsh or vice versa.

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As to why it would be a last name it could be related to location on the river or an attribute of the duck that someone once shared.